I'm in search of some order and clarity. (There might be some errors )
I'm reading a good old text most of you probably read in your youth.
I am reading about Q matrices $Q_{\psi} , Q_{\theta}, Q_{\phi}$.
It is known that an orthogonal matrix can be written in terms of a homomorphic Q matrix.
For example $Q_{\phi}$ for rotation about z is
$ \begin{pmatrix} \cos{\frac{\phi}{2} + i \sin{\frac{\phi}{2}}} & 0\\ 0 & \cos{\frac{\phi}{2} - i \sin{\frac{\phi}{2}}} \end{pmatrix} $
from
$ \begin{pmatrix} \cos{\phi} & \sin{\phi} & 0\\ - \sin{\phi} & \cos{\phi} & 0\\ 0 & 0 & 1 \end{pmatrix} $
The process for doing this is well known, and I can add it here if requested but I don't know if it will add anything useful to anyone.
$P$ and $Q$ are then derivable using the standard techniques.
I am now going to ask a question. Before I do I am now working under the assumption that this is one instance of the whole SU(2) double cover of SO(3) thing.
It seems to me that it will be possible to find a whole lot of these covering groups and to even make a claim like every group has a covering group without even thinking too hard.
For sure every lie group should. I am making these statements based on one observation.
Now that such a statement has been made, it would seem that a word proof is in order, I don't see a computational proof covering all the cases.
How does one prove or refute my statement about the existence of covering groups for every group?